Lecture sketches and homework

To help you study for this class, I am posting lecture sketches. These are not a substitute for your personal lecture notes or coming to class (for instance, lots of details and motivation are not included in the sketches). I hope that they are useful to you for revisiting the material and for preparing for exams.

After most classes, homework is assigned and posted below.

You should aim to complete the problems right after class, and before the next class meets.
A 15% penalty applies if homework is submitted after the posted due date.

Homework is submitted online, and you have an unlimited number of attempts. Only the best score is used for your grade.
Most problems have a random component (which allows you to continue practicing throughout the semester without putting your scores at risk).

Collect a bonus point for each mathematical typo you find in the lecture notes (that is not yet fixed online), or by reporting mistakes in the homework system. Each bonus point is worth 1% towards a midterm exam.
The homework system is written by myself in the hope that you find it beneficial. Please help make it as useful as possible by letting me know about any issues!

Sage

For more involved calculations, we will explore the open-source free computer algebra system Sage.

If you just want to run a handful quick computations (without saving your work), you can use the text box below.

An easy way to use Sage more seriously is by creating an account at http://cocalc.com. This free cloud service does not require you to install anything, and you can access your files and computations from any computer as long as you have internet. To do computations, once you are logged in and inside a project, you will need to create a "Sage notebook" as a new file.

Exams and practice material

There will be two in-class midterm exams and a comprehensive final exam. Notes, books, calculators or computers are not allowed during any of the exams. The exam schedule is posted in the syllabus and at the top of this page.

Quizzes and solutions

Projects

If you take this class for graduate credit, you need to complete a project. The idea is to gain additional insight into a topic that you are particularly interested in. Some suggestions for projects are listed further below.

The outcome of the project should be a short paper (about 4 pages)

in which you introduce the topic, and then

describe how you explored the topic.

Here, exploring can mean (but is not limited to)

computations or visualizations you did in, say, Sage,

working out representative examples, or

combining different sources to get an overall picture.

Let me know before spring break which topic you choose for your project.

Each project should have either a computational part (this is a great chance to play with Sage!) or have a more mathematical component. Here are some ideas:

Compute and investigate the number of Fermat liars and/or strong liars. For instance, a theoretical result states that at most a quarter of the residues can be strong liars. What proportions do you observe numerically?

Using frequency analysis (letters, digrams, trigrams and such), can you (more or less automatically) distinguish, say, different languages or maybe even individual authors. This would be a computational project. The exact focus is up to you.

What are the periods of LFSRs and LCGs? When are they maximal? Discuss mathematical results in the spirit of what is hinted at in Example 51 (for LCGs) and Example 59 (for LFSRs).

When we say that a pseudorandom generator should have good statistical properties, what exactly do we mean? What tests do people apply in practice to evaluate pseudorandom generators?

Go into more detail on the prime number theorem. How is it related to the Riemann zeta function and the Riemann hypothesis (this is advanced math)? What goes into its proof? Explore it numerically.

Discuss finite fields and their classification. This would be a more mathematical project and should include proving basic results on finite fields.

Discuss Frobenius pseudoprimes, which feature in a 1998 primality test by Jon Grantham. You could either include mathematical details, such as proofs, or implement the primality test and experimentally analyze the failure rate.

You are also very welcome to come up with your own ideas for a project. Please talk to me early with your idea, so I can give you a green light.